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Gagliardo-Nirenberg inequalities in logarithmic spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2006)

Colloquium Mathematicae

We obtain interpolation inequalities for derivatives: ∫ M q , α ( | ∇ f ( x ) | ) d x ≤ C [ ∫ M p , β ( Φ ₁ ( x , | f | , | ∇ ( 2 ) f | ) ) d x + ∫ M r , γ ( Φ ₂ ( x , | f | , | ∇ ( 2 ) f | ) ) d x ] , and their counterparts expressed in Orlicz norms: ||∇f||²(q,α) ≤ C||Φ₁(x,|f|,|∇(2)f|)||(p,β) ||Φ₂(x,|f|,|∇(2)f|)||(r,γ) , where | | · | | ( s , κ ) is the Orlicz norm relative to the function M s , κ ( t ) = t s ( l n ( 2 + t ) ) κ . The parameters p,q,r,α,β,γ and the Carathéodory functions Φ₁,Φ₂ are supposed to satisfy certain consistency conditions. Some of the classical Gagliardo-Nirenberg inequalities follow as a special case. Gagliardo-Nirenberg inequalities in logarithmic spaces with higher...

Gagliardo-Nirenberg inequalities in weighted Orlicz spaces

Agnieszka Kałamajska, Katarzyna Pietruska-Pałuba (2006)

Studia Mathematica

We derive inequalities of Gagliardo-Nirenberg type in weighted Orlicz spaces on ℝⁿ, for maximal functions of derivatives and for the derivatives themselves. This is done by an application of pointwise interpolation inequalities obtained previously by the first author and of Muckenhoupt-Bloom-Kerman-type theorems for maximal functions.

General uniform approximation theory by multivariate singular integral operators

George A. Anastassiou (2012)

Annales Polonici Mathematici

We study the uniform approximation properties of general multivariate singular integral operators on ℝ N , N ≥ 1. We establish their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate higher order modulus of smoothness. We list the multivariate Picard, Gauss-Weierstrass, Poisson-Cauchy and trigonometric singular integral operators to which this theory can be applied directly.

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